# Are injective Omega-spectra the S-local objects of symmetric spectra for some class S?

I am trying to read the Hovey-Shipley-Smith article as defining the stable model structure on symmetric spectra as a left Bousfield localization (as explained on nLab) of the projective level model structure on symmetric spectra, which has level equivalences as weak equivalences.

Almost all ingredients are there in the article. All I have left to show is that the injective Omega-spectra are indeed the S-local objects, where S is the class of stable equivalences. By defintion any map in S induces a weak equivalence of simplicial hom-sets $Map_{Sp^\Sigma}(f,E)$ for E a injective Omega-spectrum. Conversely, lemma 3.1.5 and example 3.1.10 conspire to tell you that if a symmetric spectrum is S-local and injective, it is an Omega-spectrum. So, what remains: is any S-local symmetric spectrum injective?

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You said "...where S is the set of level equivalences." Do you mean S is the set of stable equivalences? –  Tyler Lawson Dec 9 '09 at 13:14
Yes, in fact example 3.1.10 doesn't work when you define S to be the class of level equivalences, because the lambda used there is not a pi_*-isomorphism, but all level equivalences are. –  skupers Dec 9 '09 at 13:28

You have to realize it has been a long time since we wrote that paper. But I'll give it my best shot.

I think we intentionally chose the injective Omega-spectra because they are "extra fibrant", so to speak. That is, I think S-local spectra don't have to be injective, just Omega-spectra.

The injective Omega-spectra should be the fibrant objects in a different model structure. There should be an injective level structure, which I guess we did not construct, where the cofibrations are monomorphisms and the weak equivalences are level equivalences. The fibrant objects would then be the injective spectra. The injective Omega-spectra are then the fibrant objects in the left Bousfield localization of this category with respect to the stable equivalences.

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Stefan Schwede's "An unititled book project about symmetric spectra" covers, in chapter III, the projective levelwise and stable model structures on symmetric spectra in quite some detail (along with various other model structures). Specifically, Theorem 2.2 on page 131 implies that in the projective stable model structure (as a localization of the projective levelwise structure) the fibrant objects are only Ω-spectra in the "up-to-homotopy" sense.

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I think Mark Hovey has pointed out the remark necessary to finish the proof. If we work in the injective model structure, then being fibrant is equivalent to being injective. If S are the stable equivalences then the S-local objects are necessarily injective spectra. Now use lemma 3.1.5 and a generalization of example 3.1.10 to proof that injective Omega-spectra are all S-local objects. Because we changed our cofibrations, we don't exactly get the stable model structure, but we get one with the right weak equivalences and that's what counts.

I therefore think we can conclude: there is a model structure on symmetric spectra with stable equivalences as weak equivalences, which is the left Bousfield localization of the injective model structure on symmetric spectra with respect to the stable equivalences.

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See however the remark after 5.3.8 in Hovey/Shipley/Smith: Apparently this injective stable model structure is not monoidal - and it is the whole point of symmetric spectra to get a monoidal model structure... –  Peter Arndt Feb 19 '10 at 4:08